Chapter 16: Random Variables AP Statistics

Vocabulary Random Variable: A variable whose value is based on the outcome of a random event. Use capital letter, X Discrete Random Variable: A random variable that can take one of a countable number of distinct outcomes Continuous Random Variable: A random variable that can take any numeric value within a range of values. The range may be infinite or bounded at either or both ends.

There is a free game at a carnival that you attend There is a free game at a carnival that you attend. In that game, a coin is tossed twice. If the result is the same outcome (HH or TT) you win $20. If the result is first a heads and then a tails, you win $30; but if the result is first a tails and then a heads, you lose $60. What is the expected amount you’ll win in the long run?

Expected Value of Random Discrete Variable The theoretical long-run average value, the center of its model. It is a theoretical mean.

There is another game at a the carnival. This game costs $5 to play There is another game at a the carnival. This game costs $5 to play. In this game you toss a die and if a 1, 2, or is rolled, you lose (get no money); if you roll a 4 or a 5, you win $20 and if a 6 is rolled, you win $100. What is the expected amount you will win in the long run?

Obviously, if you play each game for a long stretch of time, you will win, on average, the expected value. However, there will be variation in your winnings. We can compute this variation.

Variance and Standard Deviation of Discrete Random Variable Variance: The expected value of the squared deviation from the mean Standard Deviation: Describes the spread of the data.

Find the variance and standard deviation of the two games described earlier.

Adding/Subtracting a Constant to Discrete Random Variable Suppose that for Game 2, they increase the fee to play from $5 to $8. What happens to the expected value, variance and the standard deviation?

Adding/Subtracting a Constant to Discrete Random Variable

Multiplying/Dividing a Constant to Discrete Random Variable Suppose that the payout for each outcome in Game 2 is doubled. What happens to the expected value, variance and standard deviation?

Multiplying/Dividing a Constant to Discrete Random Variable

Other “changes” What if you play the Game 1 twice (or you play once and your friend plays once)? What is the expected value, variance and standard deviation?

Other “changes” If you play both games, what is the difference in you expected values, variances, and standard deviations. We will look at the difference between Game 1 and Game 2. (Need for the outcomes of the games to be independent—or else cannot go forward with problem.)

Other “changes” with discrete (and continuous) random independent variables NOTE: You ALWAYS add variances

Continuous Random Variables Some continuous random variables have normal models (others do not). All continuous random models have means and standard deviations (and variances). If given the mean and standard deviations of a continuous random variable that can be modeled by a normal curve, we can calculate certain probabilities. When two independent continuous random variables have Normal models, so does their sum or difference.

Example It has been determined that the annual cost of medical care for dogs averages $100, with a standard deviation of $30, and for cats averages $120, with a standard deviation of $35. What is the expected difference in the cost of medical care for dogs and cats? What is the standard deviation of that difference?

Example (cont.) If the difference in cost can be described by a Normal model, what is the probability that medical expenses are higher from someone’s dog than for her cat?

Example (cont.) What if you have two dogs and a cat? What is the expected value of your annual medical costs? Standard deviation?

Example (cont) What is the probability that your annual medical bills will exceed $400?